Geometric and Algebraic Topological Methods in Quantum Mechanics

Abstract

PrefaceContemporary quantum mechanics meets an explosion of different types of quantization. Some of these quantization techniques (geometric quantization, deformation quantization, BRST quantization, noncommutative geometry, quantum groups, etc.) call into play advanced geometry and algebraic topology. These techniques possess the following main peculiarities.• Quantum theory deals with infinite-dimensional manifolds and fibre bundles as a rule.• Geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories.• Geometric and algebraic topological methods can lead to nonequivalent quantizations of a classical system corresponding to different values of topological invariants.Geometry and topology are by no means the primary scope of our book, but they provide the most effective contemporary schemes of quantization. At the same time, we present in a compact way all the necessary up to date mathematical tools to be used in studying quantum problems.Our book addresses to a wide audience of theoreticians and mathematicians, and aims to be a guide to advanced geometric and algebraic topological methods in quantum theory. Leading the reader to these frontiers, we hope to show that geometry and topology underlie many ideas in modern quantum physics. The interested reader is referred to extensive Bibliography spanning mostly the last decade. Many references we quote are duplicated in E-print arXiv (http://xxx.lanl.gov).With respect to mathematical prerequisites, the reader is expected to be familiar with the basics of differential geometry of fibre bundles. For the sake of convenience, a few relevant mathematical topics are compiled in Appendixes.

Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds

“…30 Furthermore, one can treat H (1) as an equivariant momentum mapping of Z to the Lie coalgebra G * , provided with the coordinates x i (H(z)) = H i (z), z ∈ Z. 18,31 In this case, the coinduced Poisson structure {, } N coincides with the canonical Lie-Poisson structure on G * given by the Poisson bivector field…”

Quantum mechanics with respect to different reference frames

“…[7][8][9], we quantize time-dependent mechanics in the framework of geometric quantization of the cotangent bundle T * Q with respect to its vertical polarization given by the vertical tangent bundle V T * Q of T * Q → Q. Note that polarization of T * Q need not induce polarization of V * Q, unless it contains the vertical cotangent bundle V ζ T * Q of the fiber bundle ζ (8) spanned by vectors ∂ p .…”

“…9 Otherwise, there are nonequivalent quantizations. For the sake of convenience, the compact notation (q λ , p λ ), q 0 = t, p 0 = p, is further used.…”

“…One can show that, in this case, H * is decomposed in a finite sum of products of elements of the algebra A T , though this decomposition by no means is unique. 7,9 Provided with such a decomposition, H * is an element of the universal enveloping algebra A T of the Lie algebra A T . To be more precise, it belongs to A T + A V , where A V is the universal enveloping algebra of the Lie algebra A V ⊂ A T .…”

“…7,9 The monomorphism ζ * C ∞ (V * Q) → C ∞ (T * Q) is prolonged to a monomorphism of quantum algebras A V of V * Q and A T of T * Q. These algebras consist of affine functions of momenta on V * Q and T * Q, respectively.…”